We have
$ L = \{\land , \lor , \to , \bot , \lnot \} $
and
$L_\Box = \{\land , \lor , \to , \bot , \lnot ,\Box \}$
and then we define function $\tau : L \to L_\Box$ like below
$\tau(p) = \Box p $
$\tau(\bot) = \bot $
$\tau(A \land B) = \tau(A) \land \tau(B) $
$\tau(A \lor B) = \tau(A) \lor \tau(B) $
$\tau(A \to B) = \Box( \;\tau(A) \to \tau(B)\; )$
$\tau(\lnot A) =\Box\lnot\tau(A) $
then i know that
$\tau(cl)=S5$ which cl is classical logic and
$\tau(Il)=S4$ which Il is intuitionistic logic and
$\tau(Bl)=K4$ which Bl is basic logic
and i wanna know which weaker logic than basic logic should be the input of function that output be the K logic
$\tau(???)=K$